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59
An Optimal Lower Bound on the Number of Variables for Graph Identification
 Combinatorica
, 1992
"... In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding open problem. Previously it was an open q ..."
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Cited by 135 (9 self)
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In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices. 1
Composing Mappings among Data Sources
 In VLDB
, 2003
"... Semantic mappings between data sources play a key role in several data sharing architectures. Mappings provide the relationships between data stored in different sources, and therefore enable answering queries that require data from other nodes in a data sharing network. Composing mappings is one of ..."
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Cited by 116 (7 self)
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Semantic mappings between data sources play a key role in several data sharing architectures. Mappings provide the relationships between data stored in different sources, and therefore enable answering queries that require data from other nodes in a data sharing network. Composing mappings is one of the core problems that lies at the heart of several optimization methods in data sharing networks, such as caching frequently traversed paths and redundancy analysis.
FirstOrder Logic with Two Variables and Unary Temporal Logic
 INF. COMPUT
, 1997
"... We investigate the power of firstorder logic with only two variables over ωwords and finite words, a logic denoted by FO². We prove that FO² can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", ..."
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Cited by 59 (9 self)
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We investigate the power of firstorder logic with only two variables over ωwords and finite words, a logic denoted by FO². We prove that FO² can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", and "sometime in the past", a logic we denote by unaryTL. Moreover, our translation from FO² to unaryTL converts every FO² formula to an equivalent unaryTL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the firstorder formula. We show
Describing Graphs: a FirstOrder Approach to Graph Canonization
, 1990
"... In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting ..."
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Cited by 57 (7 self)
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In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting quantifiers". We give efficient canonization algorithms for graphs characterized by Ck or Lk . It follows from known results that all trees and almost all graphs are characterized by C2 .
Temporal Logic in Information Systems
 Logics for Databases and Information Systems
, 1997
"... Temporal logic is obtained by adding temporal connectives to a logic language. Explicit references to time are hidden inside the temporal connectives. Different variants of temporal logic use different sets of such connectives. In this chapter, we survey the fundamental varieties of temporal logic a ..."
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Cited by 54 (12 self)
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Temporal logic is obtained by adding temporal connectives to a logic language. Explicit references to time are hidden inside the temporal connectives. Different variants of temporal logic use different sets of such connectives. In this chapter, we survey the fundamental varieties of temporal logic and describe their applications in information systems. Several features of temporal logic make it especially attractive as a query and integrity constraint language for temporal databases. First, because the references to time are hidden, queries and integrity constraints are formulated in an abstract, representationindependent way. Second, temporal logic is amenable to efficient implementation. Temporal logic queries can be translated to an algebraic language. Temporal logic constraints can be efficiently enforced using auxiliary stored information. More general languages, with explicit references to time, do not share these properties. Recent research has proposed various implementation t...
Conditional XPath, the first order complete XPath dialect
, 2004
"... XPath is the W3Cstandard node addressing language for XML documents. XPath is still under development and its technical aspects are intensively studied. What is missing at present is a clear characterization of the expressive power of XPath, be it either semantical or with reference to some well e ..."
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Cited by 52 (5 self)
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XPath is the W3Cstandard node addressing language for XML documents. XPath is still under development and its technical aspects are intensively studied. What is missing at present is a clear characterization of the expressive power of XPath, be it either semantical or with reference to some well established existing (logical) formalism. Core XPath (the logical core of XPath 1.0 defined by Gottlob et al.) cannot express queries with conditional paths as exemplified by "do a child step, while test is true at the resulting node." In a firstorder complete extension of Core XPath, such queries are expressible. We add conditional axis relations to Core XPath and show that the resulting language, called conditional XPath, is equally expressive as firstorder logic when interpreted on ordered trees. Both the result, the extended XPath language, and the proof are closely related to temporal logic. Specifically, while Core XPath may be viewed as a simple temporal logic, conditional XPath extends this with (counterparts of) the since and until operators.
On the Expressive Power of Datalog: Tools and a Case Study
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... We study here the language Datalog(6=), which is the query language obtained from Datalog by allowing equalities and inequalities in the bodies of the rules. We view Datalog(6=) as a fragment of an innitary logic L ! and show that L ! can be characterized in terms of certain twoperson pebble ga ..."
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Cited by 52 (9 self)
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We study here the language Datalog(6=), which is the query language obtained from Datalog by allowing equalities and inequalities in the bodies of the rules. We view Datalog(6=) as a fragment of an innitary logic L ! and show that L ! can be characterized in terms of certain twoperson pebble games. This characterization provides us with tools for investigating the expressive power of Datalog(6=). As a case study, we classify the expressibility of fixed subgraph homeomorphism queries on directed graphs. Fortune et al. [FHW80] classied the computational complexity of these queries by establishing two dichotomies, which are proper only if P 6= NP. Without using any complexitytheoretic assumptions, we show here that the two dichotomies are indeed proper in terms of expressibility in Datalog(6=).
On the Decision Problem for TwoVariable FirstOrder Logic
, 1997
"... We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity ..."
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Cited by 48 (1 self)
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We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finitemodel property, which means that if an FO²sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIMEcomplete.
Conditional XPath
 ACM Trans. Database Syst
, 2005
"... Abstract. XPath 1.0 is a variable free language designed to specify paths between nodes in XML documents. Such paths can alternatively be specified in firstorder logic. The logical abstraction of XPath 1.0, usually called Navigational or Core XPath, is not powerful enough to express every firstord ..."
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Cited by 47 (4 self)
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Abstract. XPath 1.0 is a variable free language designed to specify paths between nodes in XML documents. Such paths can alternatively be specified in firstorder logic. The logical abstraction of XPath 1.0, usually called Navigational or Core XPath, is not powerful enough to express every firstorder definable path. In this paper we show that there exists a natural expansion of Core XPath in which every firstorder definable path in XML document trees is expressible. This expansion is called Conditional XPath. It contains additional axis relations of the form (child::n[F])+, denoting the transitive closure of the path expressed by child::n[F]. The difference with XPath’s descendant::n[F] is that the path (child::n[F])+ is conditional on the fact that all nodes in between should be labeled by n and should make the predicate F true. This result can be viewed as the XPath analogue of the expressive completeness of the relational algebra with respect to firstorder logic. 1
Over Words, Two Variables Are as Powerful as One Quantifier Alternation: FO²=Sigma_2\cap Pi_2
, 1998
"... . We show a property of strings is expressible in the twovariable fragment of firstorder logic if and only if it is expressible by both a \Sigma 2 and a \Pi 2 sentence. We thereby establish: UTL = FO 2 = \Sigma 2 " \Pi 2 = UL ; where UTL stands for the string properties expressible in the tempor ..."
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Cited by 43 (8 self)
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. We show a property of strings is expressible in the twovariable fragment of firstorder logic if and only if it is expressible by both a \Sigma 2 and a \Pi 2 sentence. We thereby establish: UTL = FO 2 = \Sigma 2 " \Pi 2 = UL ; where UTL stands for the string properties expressible in the temporal logic with `eventually in the future' and `eventually in the past' as the only temporal operators and UL stands for the class of unambiguous languages. This enables us to show that the problem of determining whether or not a given temporal string property belongs to UTL is decidable (in exponential space), which settles a hitherto open problem. Our proof of \Sigma 2 " \Pi 2 = FO 2 involves a new combinatorial characterization of these two classes and introduces a new method of playing EhrenfeuchtFraiss'e games to verify identities in semigroups. While the number of variables required to express a certain graph property in firstorder logic is an important measure for the descriptional...